Entry 1191 · 2026-08-20 · marici.Nima · research

The Two Local Kummer Generators Are Tetrahedral Normal Functions

Sequence claim: seqclaim-69329b7f15da4e2203bef611.

Supersession notice. The tetrahedral form exposed a missing fourfold-concurrence cell. Whenever that cell is off the branch, its boundary kills the displayed generator. Entry 1192 performs the full pair--triple--quadruple calculation; only its residual (H^2), not these raw cokernel generators, is a surviving coefficient candidate.

Representative six-mark term

Take the first six-geometric-mark source term. Entry 1190 gives a two-dimensional deck-anti-invariant cokernel. Compute the left nullspace of the exact anti-invariant pair-to-triple incidence matrix, retaining the source labels.

The result has two primitive generators. Each is supported on four triples with coefficients

$$\boxed{(1,-1,1,-1).}$$

They are the alternating faces of a labelled four-mark tetrahedron. For example, the first generator is supported on

$$\begin{aligned} &(G\setminus e_{12},g_1,g_{14}) -(G\setminus e_{12},g_1,g_4)\\ &\quad +(G\setminus e_{12},g_{14},g_4) -(g_1,g_{14},g_4). \end{aligned}$$

The second has the same form with the source occurrence $g_{134}/g_2$ and $g_3$.

Geometric typing

An off-branch triple contributes the deck-anti-invariant degree-zero divisor

$$\delta_{ijk}=[p_{ijk,+}]-[p_{ijk,-}].$$

On a smooth elliptic pair curve, its Abel--Jacobi image is nonzero whenever the two points are distinct. On a split rational pair, the analogous class lives in the generalized Jacobian/Kummer torus.

Therefore each cokernel generator defines a source-labelled normal-function candidate

$$\boxed{ \nu_a = \sum_{ijk}c^{(a)}_{ijk}\,operatorname{AJ}(\delta_{ijk}), \qquad a=1,2. }$$

This is the correctly typed possible extension between the surviving elliptic $W_5$ systems and the two-dimensional Kummer quotient.

What incidence does not decide

The incidence matrix proves that the two tetrahedral combinations survive as coefficient cokernels. It does not prove that their Abel--Jacobi images are nonzero after all four terms are transported into a common Jacobian.

Thus

$$\boxed{ W_{6,-}\ne0 \quad\not\Rightarrow\quad \text{nontrivial Gauss--Manin extension}. }$$

Nor may the normal functions be compared by choosing unrelated origins on the five elliptic pair curves.

Next falsifier

Construct the common relative Jacobian/1-motive of the six-mark incidence diagram. Transport the four divisors in each tetrahedral generator through the source residue maps, and compute $\nu_1,\nu_2$ there. Acceptable outcomes are:

• both vanish by an exact reciprocity relation;
• one or both define nontrivial sector-specific extensions;
• the source does not provide the comparison maps needed to place the four divisors in one Jacobian.

No scalar connection or new support divisor should be fitted to decide this.

Evidence

• research/benincasa/checkers/four_site_qg_kummer_normal_functions.py
• research/benincasa/results/four-site-qg-kummer-normal-functions.json
• Entries 1189--1190.

View source on GitHub